Title: Powerful SNP Set Analysis for Case-Control Genome Wide Association Studies

Running Title: Powerful SNP Set Analysis

Michael C. Wu1, Peter Kraft2,3, Michael P. Epstein4, Deanne M. Taylor2, Stephen J.

Chanock5, David J. Hunter3, and Xihong Lin2

Department of Biostatistics1, The University of North Carolina at Chapel Hill, Chapel

Hill, NC.

Department of Biostatistics2, Harvard School of Public Health, Boston, MA.

Department of Epidemiology3, Harvard School of Public Health, Boston, MA.

Department of Human Genetics4, Emory University, Atlanta, GA.

Division of Cancer Epidemiology and Genetics5, National Cancer Institute, Bethesda,

MD.

Address for Correspondence: Xihong Lin, Ph.D.

Department of Biostatistics, Harvard School of Public Health

655 Huntington Avenue, Boston, MA 02115

Phone: (617) 432-2914

Fax: (617) 432-5619

E-mail: xlin@hsph.harvard.edu

Abstract

Genome wide association studies (GWAS) have emerged as popular tools for iden-

tifying genetic variants that are associated with disease risk. Standard analysis of a

case-control GWAS involves assessing the association between each individual geno-

typed SNP and disease risk. However, this approach suffers from limited repro-

ducibility and difficulties in detecting multi-SNP and epistatic effects. As an alter-

native analytical strategy, we propose grouping SNPs together into SNP sets based

on proximity to genomic features such as genes or haplotype blocks, and then test-

ing the joint effect of each SNP set. Testing of each SNP set proceeds via the logistic

kernel machine based test which is based on a statistical framework that allows for

flexible modeling of epistatic and nonlinear SNP effects. This flexibility as well as the

ability to naturally adjust for covariate effects are important features of our test that

make it appealing compared to individual SNP tests and existing multi-marker tests.

Using simulated data based on the International HapMap Project, we show that SNP

set testing can have improved power over standard individual SNP analysis under a

wide range of settings. In particular, we find that our approach has higher power than

individual SNP analysis when the median correlation between disease susceptibility

variant and the genotyped SNPs is moderate to high. When the correlation is low,

both individual SNP analysis and the SNP set analysis tend to have low power. We

apply SNP set analysis to analyze the CGEMS breast cancer GWAS discovery phase

data.

1 Introduction

Identification of single nucleotide polymorphisms (SNPs) that are associated with risk for

developing complex disease is an important goal of modern genetic studies. The hope is

that such knowledge can ultimately be used both for understanding the biological mech-

anisms underlying these diseases and for generating individualized risk profiles that are

useful in a public health context. To this end, genome wide association studies (GWAS)

have emerged as a popular tool for identifying common genetic variants for complex

disease. A standard case-control GWAS for identifying SNPs associated with disease sus-

ceptibility involves genotyping a large number of SNPs, on the order of hundreds of

thousands, in thousands of individuals with the disease (cases) and thousands of healthy

controls with the goal of identifying individual loci that are associated with the outcome.

Such studies have been successfully used to identify SNPs associated with susceptability

to diseases such as breast cancer1, 2 (MIM 114480), prostate cancer3–5 (MIM 176807), and

type II diabetes6–8 (MIM 125853).

A typical GWAS consists a discovery phase in which an initial set of promising sus-

ceptibility loci are identified followed by a validation stage in which the SNPs identified

in the initial discovery phase are replicated in a separate study cohort.9 The standard

approach for analyzing GWAS in the discovery phase involves individual SNP analy-

sis. This mode of analysis often involves regressing the phenotype onto each individual

typed SNP and generating a parametric p-value. The SNPs are then ranked based on their

individual p-values and a threshold is set such that all SNPs with p-value less than that

threshold will be pushed forward for validation. The threshold can be based on reach-

ing a muliple-comparison adjusted significance level or a level based on non-analytical

means.

Although use of individual SNP analysis has proved useful in identifying many dis-

1

ease susceptibility variants, this mode of analysis may be limited in some settings due to

difficulty in reaching genome wide significance. More specifically, in order to control the

overall type I error rate, the level at which each test is conducted must be adjusted. Due

to the large number of considered hypotheses, the threshold for genome wide signficance

can be very extreme and difficult to attain: for a GWAS examining the effects of 500,000

SNPs, each test is conducted at the � = 10−7 level, which is very stringent.

Additionally, individual-SNP analysis is often limited by poor reproduceability; many

of the highly-ranked SNPs in the discovery phanes are false positives and cannot be vali-

dated. This is largely due to the restricted power to detect SNPs with small effects that are

truly associated with the outcome. In particular, individual SNPs that are genotyped on

GWAS platforms often show only modest effects. One explanation for this is that the true

causal SNP is rarely genotyped, but there are typed SNPs which are in linkage disequi-

librium (LD) with the causal SNP. In this case, using individual SNP analysis, the typed

SNPs in LD with the causal SNP will each only show moderate effects since each typed

SNP serves as an imperfect surrogate for the causal SNP. Thus, it could be advantageous

to consider the joint effect of multiple SNPs in analysis10 since it is probable that several

of these markers are in LD with the causal SNP and could capture the true effect more

effectively than individual-SNP analysis. Finally, individual SNP analysis only consid-

ers the marginal effect of each SNP and therefore fails to accommodate epistatic effects.

Epistatic interactions between SNPs can contribute to disease susceptibility such that in-

dividual SNPs may show little individual effect, but their interactions can have a much

larger effect. Individual SNP analysis will not be able to detect such effects which, more

generally, are difficult to find due to the large number of potential interactions.11

As an alternative strategy for analysis, we propose grouping of SNPs together into

SNP sets along the genome and perform genome-wide tests for individual SNP sets in-

stead of individual SNPs. SNP set based analysis borrows information from different but

2

correlated SNPs that are grouped based on prior biological knowledge and hence has

the possibility of providing results with improved reproducibility and increased power,

especially when individual SNP effects are moderate, and improve interpretability of the

results. This mode of analysis proceeds via a two step procedure. First, SNPs are assigned

to SNP sets based on some meaningful biological criteria (genomic features), e.g., genes.

Then, tests for the association between each genomic feature and a disease phenotype are

performed using a logistic kernel machine based multilocus test, across the genome.

SNP set analysis can prove advantageous over the standard analysis of individual

SNPs. By forming SNP sets and testing each SNP set as a unit, we are reducing the num-

ber of hypotheses being tested and thus relaxing the stringent conditions for reaching

genome-wide significance. Grouping SNPs together properly, we will have improved

power in settings where SNPs are individually only moderately significant. In particu-

lar, though any single SNP may serve as a poor surrogate for an untyped causal SNP,

by considering multiple typed SNPs, we will be better able to capture the true effect of

the untyped causal SNP. Furthermore, if there are multiple independent causal SNPs, by

considering their joint effects, we will have power to detect their joint activity.

To test each SNP set within a case-control GWAS, we propose a general semiparamet-

ric kernel based testing procedure which is tailored towards high-dimensional genetic

data. Specifically, this test will combine the logistic kernel machine testing approach of

Liu et al.12 with the kernel framework suggested by Kwee et al.13 As we will show, the

logistic kernel machine has appealing features for SNP set analyses. The testing frame-

work is powerful and allows for great flexibility in the functional relationship between

the SNPs in a SNP set and the outcome. Thus, the method can easily account for complex

SNP interactions and nonlinear effects. Combined with the ability to seamlessly adjust

for covariate effects and the fast computational efficiency of our method, this flexibility

gives the logistic kernel machine based test significant advantages over both individual

3

SNP tests and existing multi-marker tests.

Broadly speaking, our work advances the field in three important ways. First, we de-

velop SNP set analysis as an alternative to standard individual SNP analysis and discuss

principled approaches for forming SNP sets based on genomic features. Second, we de-

velop a powerful statistical modeling and testing framework for genetic effects which has

a number of practical advantages over other multi-marker tests: our approach is compu-

tationally efficient and naturally accommodates covariate adjustment, non-linear effects,

and epistasis. Third, we will demonstrate through thorough numerical studies and data

applications that our approach can have substantially improved power over standard in-

dividual SNP testing, and by extension, over the many multi-marker tests that individual

SNP testing tends to dominate.

The remainder of this article is organized as follows. In the next section, we describe

our proposed SNP set analysis framework including how to form SNP sets and how to

subsequently test SNP sets. Then we will present simulation results comparing our ap-

proach to individual SNP analysis and two existing multi-SNP tests. Finally, we will

apply logistic kernel machine based SNP set analysis to the CGEMS breast cancer data

from the discovery phase. We will conclude with a brief discussion.

2 Materials and Methods

SNP set based analysis borrows information from different but correlated SNPs that are

grouped based on prior biological knowledge and hence provides results with improved

reproducibility and increased power, especially when individual SNP effects are moder-

ate. This mode of analysis proceeds via a two step procedure. First, across the genome,

SNPs are assigned to a SNP sets based on some meaningful biological criteria such as

proximity to genomic features—SNP sets of a single SNP are possible. If we wished to

4

perform genome wide SNP set analysis of a GWAS conducted on the Illumina Human-

Hap500 array by grouping SNPs based on genes, we could generate approximately 18000

SNP sets, each of which consisted of the SNPs within a single gene. For example, the 14

genotyped SNPs within the ASAH1 (MIM 228000) gene could be assigned to a single SNP

set and the 4 genotyped SNPs within the NAT2 (MIM 612182) gene could be assigned to

another SNP set, and so on. After the groupings are made, each of the 18000 SNP sets

is tested using a multilocus test, and the genome-wide significance of SNP set, e.g. each

gene, is calculated. Although a number of tests have been proposed,14, 15 we consider an

extension of the logistic kernel machine test, which was developed in the gene expression

profiling setting, that we tailor for analysis of genome wide association studies. In this

section, we describe possible methods for grouping SNPs in a genome wide scan into SNP

sets and then we present the logistic kernel machine test for evaluating the significance of

each SNP set.

2.1 Forming SNP Sets

A key aspect of our proposed approach is the formation of meaningful SNP sets. In prin-

ciple, a SNP set may be formed via any grouping of SNPs, and our testing approach is

still valid in the sense that the type I error rate will always be protected. However, better

groupings can be made on the basis of prior biological knowledge and if done properly,

can lead to additional gains in power. In particular, the key advantages of our approach

may be found in the ability to reduce the number of multiple comparisons, to harness

correlation between SNPs, to measure the joint effect of independent SNPs, and to make

direct inference on a biologically meaningful genomic feature. Some natural ways of

forming SNP sets that can capitalize on these advantages include grouping SNPs on the

basis of genomic features. We describe below some natural grouping structures.

A natural grouping strategy is to take all SNPs that are located in or near a gene, a

5

fundamental unit of the genome, and group them to form a SNP set. In particular, one

can take all SNPs between the start and end of transcription as well as SNPs that are

upstream and downstream of the gene, in order to capture regulatory regions, as a single

SNP set. In grouping based on known genes, we can significantly reduce the number

of multiple comparisons. The SNPs on the Illumina HumanHap 500 array correspond

to to approximately 17,800 genes in contrast to the original 530,000 SNPs. Since we take

the entire gene region, not just exonic regions, we expect to have many typed SNPs that

are correlated and thus the logistic kernel machine test will have good power to detect

a significant SNP set effect. We could also expect multiple SNPs within a gene to be

associated with disease risk and this grouping structure would allow us to detect this

effect. Testing gene-based SNP sets also makes direct inference on the association between

the gene and case-control status.

An extension of gene based SNP set analysis is to group SNPs based on whether they

are located within a gene pathway from KEGG16 or a Gene Ontology Consortium func-

tional category.17 Making inference on a pathway further reduces the number of multiple

comparisons and still allows inference on a biologically meaningful unit. The logistic ker-

nel machine test will be able to harness local LD to have power and will, additionally, be

able to capture true pathway effects when several SNPs in multiple genes are related to

the disease.

Although many variants associated with disease have been identified within gene re-

gions, many lie outside of the boundaries of known genes (and hence pathways). To

augment coverage of the genome, a possible strategy would be to group SNPs within

evolutionarily conserved regions. Increased evolutionary conservation of a genomic re-

gion is suggestive of increased importance or functionality.18 Significance of such a SNP

set would potentially indicate that there is a genomic feature present that is related to

disease risk, even if the feature is not well understood.

6

Finally, approaches to forming SNP sets that can achieve full coverage of the genome

by placing all SNPs into SNP sets include grouping SNPs via a moving window or via

haplotype blocks. For example, one could divide the genome into a fixed number of

adjacent regions, purely based on length, and treat all SNPs within a region as a SNP

set. Alternatively, one could build SNP sets based on haplotype blocks such as through

Haploview.19 Both approach will still allow us to harness local correlation to capture the

effect of untyped SNPs.

An important limitation of employing a gene or pathway based approach is the omis-

sion of intergenic regions. However, use of additional grouping strategies, e.g. conserved

regions, can augment coverage, and using the moving window and haplotybe block can

provide comprehensive coverage of the entire genome. Although we wish to group SNPs

that are near one another to harness correlation, this does not allow us to capture multi-

SNP or epistatic effects among SNPs in separate SNP sets. Using gene pathway based

SNP sets could ameliorate this issue since this looks across individual continuous regions.

Groupings based on strategies beyond the ones that we have considered are also possible.

As noted above, we emphasize that while well formed SNP sets can optimize the

power and interpretability of our SNP set testing strategy, our logistic kernel machine

testing approach is statistically valid irrespective of the grouping scheme. For illustration,

we will focus on SNP sets formed based on proximity to each of 18000 known genes.

2.2 Genome Wide SNP Set Testing

Although we propose our strategy as a genome wide approach, we will present the test-

ing procedure by focusing on testing a single SNP set.

In this paper, we assume that a population based case-control GWAS was conducted in

which n independent subjects were genotyped. To employ our SNP set analysis approach,

we first group the SNPs into SNP sets across the genome. Then for a given SNP set

7

containing p SNPs, let zi1, zi2, . . . , zip be genotype values for the SNPs in the SNP set for

the itℎ subject (i = 1,. . . ,n). The case-control status for the itℎ subject is denoted by yi

(yi = 1 for cases and yi = 0 for controls). We assume without loss of generality that

the SNPs are coded in a trinary fashion with zij = 0, 1, 2 corresponding to homozygotes

for the major allele, heterozygotes, and homozygotes for the minor allele respectively.

This corresponds to the commonly employed additive model of allelic affect, but we note

that alternative models, such as the dominant and recessive models, are also possible

and can be tested within our framework. We further assume that for each individual,

an additional set of m demographic, environmental, or other confounding variables is

collected. For the itℎ subject we let xi1, xi2, . . . , xim denote the values of the covariates we

would like to adjust for. The goal of SNP set analysis is then to test the global null of

whether any of the p SNPs are related to the outcome while adjusting for the additional

covariates.

In principle, many multi-locus testing approaches could be used for evaluating the

significance of the SNPs in the SNP set, but to harness correlation and accommodate

complex relationships between the SNPs and the outcome and epistatic effects, we pro-

pose a new approach to test the SNP set by modelling each SNP set’s effect in a flexible

fashion while adjusting for additional covariate effects. At the same time, to overcome

the issue of the large number of degrees of freedom, our strategy will employ a test that

adaptively estimates the degrees of freedom by accounting for correlation (LD) among the

SNPs. Specifically, we will choose to use the logistic kernel machine regression modelling

framework and a corresponding score test.12

2.2.1 Logistic Kernel Machine Model

In evaluating the significance of a SNP set, we need to employ a strategy that allows us

to model, and subsequently test, the effects of multiple SNPs that have been grouped

8

in a biologically meaningful fashion. The kernel machine framework has become very

popular for modelling high-dimensional biomedical data due to its ability to allow for

complex/nonlinear relationships between the dependent and independent variables20, 21

while adjusting for covariate effects. We consider a logistic kernel machine regression

model for the joint effect of the SNPs in the SNP set and the additional covariates that

we would like to adjust for. Under the notation above, for the itℎ individual, we have the

semiparametric model given by

logitP (yi = 1) = �0 + �1xi1 + ⋅ ⋅ ⋅+ �mxim + ℎ(zi1, zi2, . . . , zip) (1)

where �0 is an intercept term, �1, . . . , �m are regression coefficients corresponding to the

environmental and demographic covariates. The SNPs, zi1, . . . , zip, influence yi through

the general function ℎ(⋅) which is an arbitrary function that that has a form defined only

by a positive semidefinite kernel function K(⋅, ⋅).

Our primary aim is to adequately model the SNPs and evaluate their effect, so ℎ(⋅)

is the model component in which we have primary interest because it fully determines

the relationship between genotypes of the SNPs in the SNP set and disease risk. Leaving

ℎ(⋅) only generally specified permits a modelling framework that accommodates complex

relationships between the SNPs and risk as well as epistatic effects.

We omit the mathematical details, but using the representer theorem,22 we note that

ℎ(zi1, zi2, . . . , zip) in Equation 1 is equal to ℎi = ℎ(Zi) =∑n

i′=1 i′K(Zi,Zi′) for some

1, . . . , n. This shows that ℎ(⋅) is fully defined by the kernel function K(⋅, ⋅). Details on

the mathematical relationships and estimation may be found in Liu et al.12 and Cristianini

et al.,20 but the key is that by choosing different kernel functions, we can specify different,

possibly complex, bases and corresponding models. For example, if we define K(⋅, ⋅) to

be the linear kernel such that K(Zi,Zi′) =∑p

j=1 zijzi′j then we are implicitly assuming the

9

simple logistic model defined by

logitP (yi = 1) = �0 + �1xi1 + ⋅ ⋅ ⋅+ �mxim + �1zi1 + �2zi2 + ⋅ ⋅ ⋅+ �pzip

where �j is a regression coefficient corresponding to the jtℎ SNP. To specify a more com-

plicated model, we need only change our choice of K(⋅, ⋅).

From the above, it is apparent that the choice of kernel changes the underlying basis

for the nonparametric function governing the relationship between case-control status

and the SNPs in the SNP set. Essentially, K(⋅, ⋅) is a function that projects the genotype

data from the original space to another space and then ℎ(⋅) is modelled linearly in this new

space, such that if one considers ℎ on the original space, it can be highly nonlinear. More

intuitively, however, K(Zi,Zi′) can be viewed as a function that measures the similarity

between two individuals, the itℎ and i′tℎ subject, based on the genotypes of the SNPs in the

SNP set. Taking this perspective, many choices for K are possible. Some specific kernels

functions that we can consider include the linear, identical-by-state (IBS), and weighted

IBS kernels.

The linear kernel is: K(Zi,Zi′) =∑p

j=1 zijzi′j which is the usual inner product between

the covariate vectors for subject i and i′. As described earlier, this kernel assumes a set of

basis functions that spans the original covariate space such that one is implying a linear

relationship between the logit of the probability of being a case and the genotypes of the

SNPs in the SNP set, i.e. the usual multiple logistic regression model.

The gaussian kernel is: K(Zi,Zi′ ; d) = exp{−∑p

j=1(zij − zi′j)2/d} and assumes the

radial basis which is difficult to characterize using an explicit set of basis functions. The

class of models generated by the gaussian kernel can be very broad and includes the

linear model as a special case. Here d is a parameter that approximately controls area

of influence of the kernel function such that larger values of d correspond to smoother ℎ

10

functions.

The IBS kernel is: K(Zi,Zi′) =∑p

j=1{2I(zij=zi′j)+I(∣zij−zi′j ∣=1)}2p

. In genetics, a possible

metric for evaluating distance between individuals on the basis of genotype information

is the number of alleles shared identical by state (IBS) by a pair.15 As shown by Kwee et

al.,13 this may also be used a a valid kernel function.

The weighted IBS kernel is: K(Zi,Zi′ ;w) =∑p

j=1 wj{2I(zij=zi′j)+I(∣zij−zi′j ∣=1)}2p

where wj =

1/√qj and qj is the minor allele frequency (MAF) for the jtℎ SNP in the SNP set. The

weighted IBS kernel is an extension of the IBS kernel that up-weights for similarity in

rare alleles. The idea is that similarity in rare alleles is more informative than similarity

in common alleles.

The ability to model data using the gaussian and IBS kernels are advantages of using

the kernel machine framework since formulating an explicit set of basis functions can be

difficult. Alternative kernel functions, such as those discussed in Wei and Schaid23 and

in Mukhopadhyay et al.24 are possible and can be designed for specific data sets. To be a

valid kernel function, K(⋅, ⋅) needs to be positive semi-definite and satisfy the conditions

of Mercer’s theorem.20

2.2.2 Logistic Kernel Machine Test

Here, our focus is on hypothesis testing for which only need to estimate � under the null

hypothesis that ℎ(Zi) = 0. Therefore, we omit the technical details on estimating the

genetic effect, ℎ(Z), from the SNP set and refer the reader to Liu et al.12

The above modelling framework leads naturally to a powerful test for association

between the SNPs in the SNP set and case-control status. Note that the probability that

the itℎ subject is case depends on the SNPs only through the function ℎ(Zi). Thus, in order

11

to test whether there is a true SNP set effect, we can consider the null that

H0 : h(Z) = 0 (2)

against the general alternative. To test this hypothesis, Liu et al.12 exploit the connection

between the kernel machine framework and generalized linear mixed models (GLMM).

Specifically, letting K be the n × n matrix with (i, i′)tℎ element equal to K(Zi,Zi′), then it

is straightforward to see that h = K , where h = [ℎ1, . . . , ℎn]′. We can treat h as a subject

specific random effect, then via the GLMM connection, h follows an arbitrary distribution

F with mean zero and variance �K. Note that � indexes the effect of the SNPs in the SNP

set such that

H0 : ℎ(Z) = 0⇔ H0 : � = 0.

Thus, we need only to test whether the indexing parameter � is significantly different

than zero. This can proceed via the variance component score test of Zhang and Lin25

using the statistic:

Q =(y − p0)

′K(y − p0)

2(3)

where logit p0i = �0 + �1xi1 + �2xi2 + ⋅ ⋅ ⋅+ �mxim. Since this is a score test, �0 and the �j

are estimated under the null model which does not contain h, so we can use the standard

estimate from the logistic regression model without the genotypes. To compute a p-value

for significance, we can compareQ to a scaled �2 distribution with scale parameter, �, and

degrees of freedom, �. Details on calculating � and � are found in the Appendix.

The adaptive estimation of the degrees of freedom, �, constitutes a key advantage of

the logistic kernel machine test. In particular, if the R2 between the SNPs in the SNP

set increases, then � decreases such that if all the SNPs are perfectly correlated, � →

1. It follows that for a given ℎ(⋅), higher correlation is likely to lead to higher power,

12

suggesting that the logistic kernel machine test improves the power for SNP set testing

by harnessing the correlation between SNPs and adaptively estimating �.

In general, it can be difficult to identify a prior whether it is the minor allele or the

major allele that is associated with increased disease risk, and equivalently, whether the

minor allele is protective or deleterious. The logistic kernel machine test is not affected

by the directionality of effect and its power is robust to whether the minor alleles of the

causal SNP are protective or deleterious (or a combination of both in settings with multi-

ple causal variants).

The testing framework considered here has similarities to those of Schaid et al.,10

Mukhopadhyay et al.,24 and Wessel and Schork15 which we describe below in that all

three approaches are based on genetic distances among subjects. However, the kernel

framework allows for improved flexibility in the functional relationship.

2.2.3 Existing Multi-SNP Tests

Although other multi-SNP tests could be used for evaluating the significance of each SNP

set, the kernel machine has advantages over each of these. Here, we briefly discuss some

alternative tests that fall into several different categories.

The first class of multi-SNP test encompasses the multi-marker methods that are based

on individual SNP analysis. In particular, a common approach for evaluation the signif-

icance of a set of markers is to apply individual SNP analysis by testing the individual

significance of each SNP, using the most significant p-value as the p-value for the set of

loci, and then correct for having done multiple tests via monte carlo methods26 or by esti-

mating the effective number of tests.27–29 Alternatively, the test statistics from each of the

individual tests can be combined.30 However, such tests still rely strongly on individual

SNP analysis and when the individual SNPs are not in high LD with the causal variant,

they may have low power, as they do not borrow information across SNPs which are fre-

13

quently correlated. Furthermore, they cannot accommodate complex genetic effects and

interactions. Our simulations will verify that the logistic kernel machine test often has

improved power over this class of test.

Omnibus tests for multiple SNPs or haplotypes via multivariate regression10, 31 allow

for simultaneous analysis of all SNPs, butstudies have shown that such methods often

offer little benefit over individual SNP analysis based methods32, 33 as they are based on

a large number of degrees of freedom. To reduce the degrees of freedom, a set of multi-

marker tests that compare pairwise genetic similarity with pairwise trait similarity were

proposed by Schaid et al.,14 Wessel and Schork,15 and Mukhopadhyay et al.24 All three

approaches are attractive; however, as noted by Mukhopadyay et al., an important limi-

tation of Schaid et al.’s approach is that it assumes all variants have the same direction of

effect, i.e. all the minor alleles for each SNP increase risk or all minor alleles decrease risk.

Although the methods of Wessel and Schork and Mukhopadhyay et al. are robust to di-

rectionality, both evaluate significance via computationally expensive permutation which

may be impractical for some GWAS settings. None of the three similarity based meth-

ods allow for easy covariate adjustment. The logistic kernel machine test also considers

pairwise similarity and shares the attractive nonparametric SNP effects model, but in ad-

dition to using a computationally efficient score test and being robust to directionality,

the logistic kernel machine model naturally incorporates covariate effects, an important

feature. Beyond adjusting for confounders and population structure, it is often neces-

sary to adjust for highly significant SNPs in GWAS to distinguish between settings where

a particular significant marker is the causal SNP (or a SNP in high LD with the causal

SNP), versus setting where additional independent markers that are associated with dis-

ease are present. A third similarity based approach by Tzeng and Zhang34 can be seen

as a special case of the more general logistic kernel machine test that focuses exclusively

on haplotype similarity. The need to phase sample haplotypes from genotype data incurs

14

additional computational expense and variability—particularly for larger SNP sets.

A final class of multi-marker tests consists of methods that leverage explicit popula-

tion genetic models to pinpoint the causal locus. Many involve reconstructing the sample

phylogeny to guide the analysis and infer the causal mutation.35, 36 If the population ge-

netics model assumed is realistic and correct, such problem specific methods should have

high power. However, it is difficult to validate the assumed models and most proce-

dures are computationally intensive such that in real applications the models need to be

simplified. Once again, these models usually fail to allow for covariate adjustment. Com-

putational efficiency and ease of covariate adjustment give a practical advantage to the

logistic kernel machine regression test.

2.3 Simulations

To evaluate the performance of our SNP set analysis approach, we study the logistic ker-

nel machine test in the genetics framework by considering its empirical performance un-

der a variety of settings. For simplicity of implementation, all causal SNPs in our simu-

lations are assumed to increase disease risk, but it is important to note that none of the

methods we consider are affected by the direction of effect.

2.3.1 Simulations Based on the ASAH1 Gene

We first investigate the size and power of the kernel machine testing framework under a

setting in which the SNP set is generated based on the LD structure of a single gene which

will allow us to better understand under which settings our SNP set analysis approach is

most advantageous. We considered the ASAH1, NAT2, and FGFR2 (MIM 176943) genes,

but for clarity, we present only the simulation configurations and the results based on

the ASAH1 gene. The simulations and results from using the NAT2 and FGFR2 were

qualitatively similar.

15

ASAH1, acid ceramidase 1, is a 28.5kb long gene with 86 HAPMAP SNPs and is lo-

cated at 8p21.3-p22. Expression is associated with prostate cancer37 and mutations in the

gene are known to be associated with Farbers Disease38 (MIM 228000). We based our gene

specific simulations on the LD structure of the ASAH1 gene and used HAPGEN39 and the

CEU sample of the International HapMap Project40 to generate SNP genotype data at each

of the 86 loci.41 14 out of 86 SNPs are genotyped using the Illumina HumanHap500 array.

These will be the “typed” SNPs we use for our simulated analysis.

We first conducted simulations to verify that the logistic kernel machine test properly

controls the type I error rate. To investigate the empirical size of our test, we conducted

simulations in which we generated n/2 cases and n/2 controls under the null logistic

model where disease risk does not depend on the genotype:

logitP (yi = 1∣Xi) = �0 + �′Xi (4)

where Xi is a vector of additional covariates that are independent of the simulated geno-

type data. We considered n = 1000, 2000 and also considered the use of the linear, IBS,

and weighted IBS kernels. For each choice of n and kernel function we generated 5000

data sets using HAPGEN. To ensure that our simulations are realistic, our simulations

generated all 86 HapMap SNPs, but we only apply our testing approach to the 14 typed

SNPs. Specifically, we group the 14 SNPs as a SNP set based on the ASAH1 gene and

then we apply the logistic kernel machine test to compute a p-value evaluating the effect

of the SNPs in the SNP set while adjusting for covariates in X. For comparison, we also

analyzed the 14 typed SNPs as we would have done under an individual SNP analysis:

we tested the significance of each of the 14 SNPs individually, while again adjusting for

covariates in X, and then adjusted the individual p-values via a modified bonferroni cor-

rection where the effective number of tests was computed via two approaches. First, we

16

used the method of Moskvina et al.;29 second, we estimated the effective number of tests

as the number of principal components necessary to account for 99% of the variability.42

The two approaches were approximately concordant. The smallest p-value, corrected for

the effective number of tests, was was taken as the p-value for the entire SNP set. Size for

individual SNP analysis testing was again the proportion of p-values less than � = 0.05.

To compute the empirical power for a SNP set, we generated data sets with n/2 cases

and n/2 controls under the alternative logistic model:

logitP (yi = 1∣Xi) = �0 + �′Xi + �czci (5)

where zci is the genotype for the “causal” SNP, �c is the log genetic odds ratio for the

causal SNP, and Xi are a vector of additional covariates that are independent of zci . Note

that under each simulation configuration we allow only a single causal SNP. Each of the 86

HapMap SNPs was set to be the “causal” SNP in turn. Setting �c = 0.2 which corresponds

to a genetic odds-ratio of 1.22, we again considered sample sizes n = 1000, 2000. For

each choice of n, and for each of the 86 causal SNPs, we generated 2000 data sets. We

again apply our testing approach to each data set by grouping the 14 typed SNPs and

computing a p-value for the significance of the SNP set, while adjusting for covariates in

X, via the logistic kernel machine test under a linear kernel. We emphasize that only the

14 typed SNPs were used so the causal SNP is unobserved under most configurations.

For each configuration, we then computed the test power as the proportion of p-values

less than the � level = 0.05. This was compared with the power based on the individual

SNP analysis with modified bonferroni correction approach described above.

17

2.3.2 Simulations Based on Randomly Sampled Genes

We also evaluate the power of our approach under settings in which the LD structure

of the simulated SNP sets varied across a wide range of possible genes. Specifically, we

generated 20,000 SNP sets using HAPGEN where each SNP set is based on a real gene on

chromosome 10. This allows for 670 possible SNP sets. Within each SNP set, we randomly

selected one HapMap SNP to be the causal SNP and again generated n/2 cases and n/2

controls based on model given by Equation 5 with �c again fixed at 0.2 (OR = 1.22). Again

treating the SNPs on the Illumina HumanHap 500K array as the typed, we tested the

significance of the SNP set using the logistic kernel machine test under a linear kernel.

We also apply the individual SNP analysis testing procedure described above. Thus, for

both our method and the competing individual SNP analysis test, we computed 20,000

p-values for significance.

2.3.3 Comparisons with Alternative Multi-SNP Tests

As discussed previously, in principle, any multi-SNP test can be used to test the signifi-

cance of a SNP set. However, the kernel machine test is advantageous in that it adaptively

finds the degrees of freedom of the test statistic in order to account for LD between geno-

typed markers, can permit complex relationships between the SNPs and the outcome,

naturally allows for covariate adjustment, and is computationally efficient since no per-

mutation is required. To provide additional empirical results, we compare the logistic

kernel machine test to the similarity based testing approach of Mukhopadhyay et al24

and the approach of Wessel and Schork,15 which has been found to perform well relative

to other multi-SNP tests.23 We assessed the power under five models and the test size

under two additional models. For each of the five models examining power, 500 sim-

ulations were conducted, and 1000 simulations were conducted under the two models

18

examining the test size. For all seven models, we assumed sample sizes of 500 case and

500 controls, 1000 permutations were used to compute the p-values for the methods of

Wessel and Schork and Mukhopadhyay et al., and power and size were computed as the

proportion of p-values less than 0.05.

We first compare the power of the methods under four alternative models using SNP

sets based on the ASAH1 gene. Under Model 1, the data sets were simulated under the

alternative logistic model based on Equation 5 in which the causal variant was fixed to

be rs3810 (the third in the SNP set), one of the 14 typed SNPs, with �c again fixed at

0.2 (OR = 1.22). Model 2 was similar to Model 1, except we change the causal SNP to

be rs7825389 (the 69tℎ SNP), an untyped SNP. Model 3 was again similar to the earlier

models, except we allow for two causal variants, rs10105871 and rs7825389, which are the

63rd and 69tℎ SNPs in the ASAH1 SNP set, respectively. Both SNPs are in the same LD

block and rs10105871 is typed while rs7825389 is untyped. The effect size for both causal

SNPs was set at 0.2. Model 4 is identical to Model 3, but here we have two untyped causal

SNPs which are in different LD blocks, rs4377998 and rs7825389, which are the 43rd and

69tℎ SNPs respectively.

Under Model 5, we compared the power under the setting considered by Mukhopad-

hyay et al. in which 10 independent markers in Hardy-Weinberg equilibirium with MAF

= 0.05 are simulated. Two of the ten markers were causal with relative risks of 1.25 under

an additive model, and all 10 markers were considered to be genotyped. No additional

covariates are present.

We compare the type I error rate control of the logistic kernel machine test and the

approaches by Wessel and Schork and Mukhopadhyay et al. Specifically, under Model 6,

we simulated null data sets based on Equation 2 and the ASAH1 gene. We applied both

approaches to each of the data sets to estimate p-values for the significance of the SNPs

in the SNP set, and the size at the for each approach was estimated as the proportion of

19

p-values less than the 0.05 significance level. Under Model 7 is similar to Model 6, but we

generate an additional demographic covariate that is correlated with rs3810 (� = 0.065),

the third SNP in the SNP set.

2.4 CGEMS Breast Cancer Data

To demonstrate the applicability and power of our approach on real data, we apply SNP

set analysis to real GWAS data and contrast our results with those found under individual

SNP analysis.

The Cancer Genetic Markers of Susceptibility (CGEMS) breast cancer study1 was con-

ducted to identify individual SNPs associated with breast cancer risk. To this end, in the

discovery phase, 1,145 cases with invasive breast cancer and 1,142 controls were geno-

typed at 528,173 loci using an Illumina HumanHap500 Array. All subjects were post-

menopausal women of European ancestry recruited from the Nurses Health Study. The

results of the top SNPs from the discovery phase are given in Table 2. In the initial vali-

dation study, the top 6 SNPs as well as two others in the FGFR2 gene were genotyped in

an independent set of 1,776 cases and 2,072 controls. A SNP within FGFR2 was validated

and found to be associated with risk of breast cancer. Note that the SNPs in FGFR2 were

not the top ranked variants and that the variants within FGFR2 do not reach genome wide

significance using either the bonferroni correction or an FDR correction in the initial scan.

To evaluate the performance of SNP set analysis with the logistic kernel machine based

test by applying it to reanalyze the CGEMs Breast Cancer Data. Specifically, we formed

SNP sets by grouping SNPs that lie within the same gene. To ensure that SNPs with

possible gene regulatory roles were also included in the SNP sets, all SNPs from 20kb

upstream of a gene to 20kb downstream of a gene were grouped. Using these criteria we

were able to assemble a total of 17,774 SNP sets that consisted of 310,219 unique typed

SNPs. We tested each of the gene based SNP sets using the logistic kernel machine test

20

under the linear kernel, the IBS kernel, and the weighted IBS kernel. SNPs were coded in

the additive mode and we adjusted for parametric effects of age group, whether the indi-

vidual had hormone therapy, and the first four principal components of genetic variation

to control for population stratification.43

3 Results

3.1 Empirical Size and Power Based on the ASAH1 Gene

The size results for the logistic kernel machine test and individual SNP analysis are pre-

sented in Table 1. Based on our simulations, the logistic kernel machine test has correct

size for the kernels and sample sizes corrected and therefore, our overall strategy of logis-

tic kernel machine based SNP set analysis protects the type I error rate. Individual SNP

analysis with modified bonferroni correction also has correct size. As expected, the av-

erage effective number of tests over the 5000 replicates was stable irrespective of sample

size: 8.22 for n = 1000 and 8.23 for n = 2000.

We present the empirical power results for simulation based on the ASAH1 gene in

the top panel of Figure 1. The power for each testing approach and sample size is shown

for each of the 86 HapMap SNPs acting as the causal SNP. Based on Figure 1, we can

see that both methods have power when the causal SNP is in moderate or high LD with

the 14 typed SNPs. In these settings, the power for our logistic kernel machine SNP set

analysis approach tends to dominate individual SNP analysis for both considered sample

sizes suggesting that our testing approach is an attractive alternative or auxiliary method

to individual SNP analysis. For settings in which the causal SNP was not in LD with the

typed SNPs, the power was approximately at the type I error rate as we would expect.

For the purpose of clarifying the optimal conditions for our testing approach, Figure

21

2 shows the power for each testing approach and sample size is again presented, but here

the causal SNPs on the horizontal axis are ordered by the median R2 of the causal SNP

with the 14 typed SNPs. The median R2 between the causal SNP and the 14 typed SNPs

is plotted in the bottom panel. It is evident from the plots that the power for both testing

approaches grows as a function of the median R2 between the causal SNP and the typed

SNPs. On the right side of the plot where the median R2 is moderate to high, the ker-

nel machine based testing tends to have dramatically improved power over individual

SNP analysis even when the causal SNP is genotyped. When the median R2 is low, nei-

ther approach has much power. We emphasize that we consider the median R2 and not

the maximum and note that the power for the kernel machine test is not necessarily the

highest for situations in which the causal SNP is typed.

We repeated the size and power calculations based on the ASAH1 gene for SNPs coded

in a dominant model (results not shown). We also repeated power calculations for SNP

sets with LD structure based on the FGFR2 and NAT2 genes. The size was again correct

and power plots are qualitatively similar.

The empirical studies show that logistic kernel machine based SNP set analysis pro-

tects the type I error rate. Furthermore, except for SNPs in low LD with the genotyped

SNPs (for which neither method has any power beyond the type I error rate and hence

any differences in power are random), the kernel machine based SNP set analysis has

greater power than individual SNP analysis.

3.2 Empirical Power Based on Randomly Sampled Genes

To summarize our results, we divide the 20,000 simulations into three groups based on p,

the number of typed SNPs with the SNP set. Essentially, we compute power after binning

the 20,000 simulations based on the SNP set size and then the median R2 between the

causal SNP and the typed SNPs. More specifically, we split the simulations in groups

22

where p ≤ 10, where 10 < p ≤ 20, and where 20 < p. Then we further divided each of the

three groups into subgroups by sorting the simulated SNP sets based on the median R2

between the causal SNP and the typed SNPs and then splitting the group into 50 evenly

sized subgroups. Within each subgroup, we estimated the power as the proportion of

p-values less than � = 0.05. For each of the groups, we plot the kernel density smoothed

power against the median R2 for the subgroups in Figure 3. We need to divide the SNP

sets based on the number of SNPs because distantly located SNPs are uncorrelated such

that the median R2 decreases with increased numbers of typed SNPs.

The plots verify the earlier result we found that the power increases as a function of the

medianR2 between the causal SNP and the typed SNPs. If the causal SNP is uncorrelated

with most typed SNPs then we have little power to detect the SNP set effect, but if there

is any power, then the kernel machine based SNP set analysis method again tends to

have higher power than individual SNP analysis. Both the overall power and the relative

power of our approach to individual SNP analysis increases as the number of typed SNP

increases. This again indicates that our approach may be a better alternative to individual

SNP analysis.

3.3 Multi-SNP Test Comparison Results

The results comparing the power and type I error rates of the logistic kernel machine test

and the Wessel and Schork approach are presented in Figure 4. As expected, if the num-

ber of independent causal SNPs is increased, the power for both approaches increases.

Across the first 4 models which compare the empirical power under practical settings

based on the ASAH1 gene, the logistic kernel machine test tends to have higher power

than both the Wessel and Schork method, with a gain of approximately 12-18%, and the

approach of Mukhopadhyay et al., which improves little over the type I error rate. Under

Model 5, which assumes common MAF and no LD among typed SNPs within a gene and

23

2 causal SNPs that are genotyped, the logistic kernel machine test and Mukhopadhyay et

al.’s approach perform similarly, and both have considerably higher power than the Wes-

sel and Schork method. Overall, these results suggest that the logistic kernel machine test

has optimal power relative to other multi-SNP tests across different patterns of LD. More

interesting are the simulations comparing the type I error rate. When the demographic

and environmental covariates were simulated independently of the genotype informa-

tion, the size for all three tests is correct. However, when we set correlation between the

covariates, which is associated with the outcome and the genotypes to be modest (0.065),

failing to account for the covariates using the Wessel and Schork and Mukhopadhyay et

al. methods can possibly lead to an apparently inflated type I error rate of 25% and 10%,

respectively. This illustrates the importance of evaluating the significance of SNP sets

while in the presence of possible confounders.

Additional power simulations based on the ASAH1 gene in which as many as 4 causal

SNPs were used did not yield qualitatively different results in that the logistic kernel

machine test tended to have higher power. As this is unlikely to be a realistic situation,

given the rarity of risk-associated common variants and the relatively small regions, these

results are omitted.

We note that Mukhopadhyay et al.’s approach has similar power to the logistic kernel

machine test under Model 5. This is a setting that is favors their approach. In partic-

ular, the method of Mukhopadhyay et al. is based on an ANOVA model that assumes

that the effects of the modeled SNPs are constant and the residual correlation among ker-

nel similarity scores is the same across all different pairs of cases or controls considered.

Consequently, the method of Mukhopadhyay et al. will have excellent power when these

modeling assumptions hold but may lose power when such assumptions are violated,

such as under Models 1 through 4. The logistic kernel machine test does not make the

same assumptions as the method of Mukhopadhyay et al.; for example, the effect sizes of

24

the modeled SNPs and MAFs are allowed to vary in our approach.

Since the power of the logistic kernel machine tends to be comparable or higher, and

given the difficulties posed by failing to adjust for demographic and environmental co-

variates and the additional computation cost incurred by permutation, the logistic kernel

machine test appears to be an attractive approach for testing the significance of SNP sets.

3.4 CGEMs Breast Cancer Data Analysis Results

The results of our reanalysis may be found in Table 3. Using our approach and the linear

kernel, we see that the SNP set formed of genetic variants close to the FGFR2 is now the

most highly ranked SNP set with p-value equal to 7.69×10−7 and FDR q-value equal to

0.01. At that signficance level, it also reaches genome wide significance if we apply a

bonferroni correction (� = 0.05/17, 774 = 2.8 × 10−6) or if we control the false discovery

rate. Using a bonferroni correction, FGFR2 again reaches genome wide significance if we

apply use the IBS kernel, and if we control the FDR at 5% it reaches significance with the

weighted IBS kernel as well.

4 Discussion

In this article, we propose logistic kernel machine based SNP set analysis as an approach

for the analysis of case-control genome wide association studies. Our approach employs

prior biological knowledge to group multiple SNPs that are located near genomic features

into SNP sets and then tested as a single unit. Specifically, we choose to model the SNPs

in the SNP set using a flexible semiparametric modelling framework which is based on

kernel machines and we choose to test the effects of the SNP set via a powerful variance

components test. We illustrate our approach using both data simulated from the Inter-

national HapMap Project40 as well as the CGEMS Breast Cancer GWAS study of Hunter

25

et al.1 and showed that our approach is an attractive alternative or auxiliary approach to

individual SNP analysis.

The logic behind our analysis strategy is that we can borrow information between

different SNPs to improve power to detect true effects. Thus the choice of grouping can

influence the power of our approach. We focused on grouping SNPs based on their prox-

imity to a known gene and noted that this allowed us to reduce multiple comparisons

and harness local LD structure to improve power to capture untyped SNPs. Using genes

as the genomic features of interest allows us to map approximately 310K SNPs to 18K

SNP sets. However, it may be that the causal SNP lies far from a known gene in which

case groupings based on genes (and pathways by extension) will fail to capture the effect

of interest. To augment coverage of gene desert regions, we can group SNPs based on

additional genomic features such as evolutionarily conserved regions. Such groupings

again allow us to harness local correlation. The moving window approach will be useful

for capturing all genotyped SNPs, but direct interpretation of SNP set analysis results are

more difficult, though this may not be important. Groupings via haplotype blocks are at-

tractive since they make explicit use of the LD information. Use of haplotype blocks will

allow for comprehensive coverage of the entire genome and remove the need to explicitly

predefine genomic features of interest.

Beyond harnessing local LD structure to boost power, another important feature of our

approach is the ability to model the joint effect of multiple independent causal signals

as well as possible epistatic effects. Practically, however, finding a SNP set formation

strategy that optimizes for this can be difficult. Using a gene or moving window strategy

can certainly capture multi-SNP and epistatic effects among SNPs that are located close

to one another on the genome, but identification of such signals among SNPs that are

distantly placed will not be possible. A potential strategy is to use existing prior biological

knowledge. In particular, if multiple SNPs are expected to affect the disease risk, it is

26

not unreasonable to expect them to lie within genes in the same pathway or genes with

similar function; hence, forming SNP sets based on pathways can potentially capture such

effects. Unfortunately, a systematic approach for identifying such grouping structures

at the genome wide level is not obvious. To avoid bias in our testing procedure, any

grouping strategy must be made without consideration of the case-control status of the

subjects in the data set. Thus, groupings must be made using information from external

sources, prior studies, or unsupervised statistical methods. As such, SNP set formation

strategies will improve with advances in our knowledge of the genome and genomic

structures.

Although we focused our power simulations on the linear kernel, our simulation re-

sults nevertheless suggest that our approach is as powerful as individual SNP analysis

and our approach can often have improved power over both the individual SNP analysis

strategy and other multi-SNP testing methods. In particular, we are able to show that

when the causal SNP is correlated with multiple typed SNPs, our approach has higher

power than individual SNP analysis. In settings where the causal SNP is not correlated

with multiple typed SNPs, simulations show neither individual SNP analysis nor our ap-

proach will be able to detect an effect. Recall that, here, the term individual SNP analysis

refers to correcting the smallest individual p-value for the SNPs in the SNP set for multi-

ple comparisons and using the adjusted p-value as the p-value for the entire SNP set. The

minimum uncorrected p-value for a SNP set may be smaller than the p-value from the lo-

gistic kernel machine test but would lead to significantly inflated type I error rate. Under

several settings, we found the kernel machine test tended to have improved power over

competing multi-SNP tests while naturally allowing for covariate adjustment to protect

the type I error rate when confounders are present.

We noted earlier that the linear kernel corresponds to the usual simple logistic model

whereas the IBS and weighted IBS are kernels tailored specifically to genetic data and the

27

quadratic kernel is potentially useful for modelling epistatic effects. In fact, when epistatic

effects are present, the IBS kernel can allow for dramatically improved power over the

linear kernel. The ability to allow for complex relationships between the SNPs by just

specifying a single distance metric is an attractive feature of our approach. In practice,

however, one needs to choose a kernel a priori. Although our simulations demonstrated

that the size of our test is correct irrespective of the kernel used, the power will be in-

fluenced by the choice of kernel. The best way to choosing a kernel to use is unclear

since methods using the data to be tested are likely to overfit and simulations may reflect

the process under which the data were simulated. Our experience in simulations and

real data applications suggests using the linear kernel for testing SNP sets in which no

epistatic effects are anticipated (such SNP sets based on short regions) and the IBS kernel,

otherwise. Our experience is that there is a small loss in power for using the IBS kernel

when the true effect is linear, but potentially a considerable loss in power when the true

effect is complex/epistatic and the linear kernel is applied. Future research is necessary

to study the power using other types of kernels.

Our numerical results lead us to recommend our kernel machine approach for per-

forming multi-SNP analysis across a range of realistic settings. We have shown that it has

more power compared to existing popular approaches. It also has the ability to adjust for

covariates. This is particularly attractive since one usually needs to control for possible

population stratification and additional confounders in association studies. As noted by

Mukhopadhyay et al.,24 the performance of individual multi-SNP tests can depend on a

range of factors including the number of causal SNPs, effect size, and LD structure. Fu-

ture research is needed for more comprehensive comparisons, e.g. in other settings and

with other multi-SNP methods.

For a SNP set that is significantly associated with disease susceptibility, it is of great in-

terest to subsequently perform fine mapping and identify the individual causal variants.

28

One strategy that can be used is to apply a variable selection procedure to select the “most

important” SNPs. For instance, one could use a LASSO penalized logistic regression44 to

regress the case-control status on the 14 SNPs in the ASAH1 SNP set. LASSO penalized

logistic regression will cause some of the regression coefficients to be estimated as ex-

actly zero, dropping the corresponding variables from the model. Such a strategy has

been used by others.45–47 However, existing variable selection literature does not allow

for selection of features within the logistic kernel machine regression framework in the

presence of SNP-SNP interactions. The optimal strategy for quantifying the contributions

of individual SNPs remains an area of considerable interest.

In addition to being able to account for complex SNP effects and adjust for covariates,

the key advantage of the logistic kernel machine test is the ability to adaptively estimate

the degrees of freedom. As discussed earlier, when the genotyped SNPs are highly cor-

related, the degrees of freedom of the test remain approximately constant. As a result,

the strength of our method can increase as progress in genotyping technology allows for

denser screens.

Appendix

Approximating the Null Distribution of the Score Statistic for the Logis-

tic Kernel Machine Test

The score statistic Q defined by Equation 3 tests the null hypothesis that H0 : � = 0

and is based on the variance components tests developed by Zhang and Lin48 and Lin49

and adapted by Liu et al.12 Note that this is a boundary case, so the null distribution

for Q follows a complex mixture of �2. This can be approximated via the Satterthwaite

method50 as a scaled chi-squared distribution, ��2� , where the scale parameter, �, and

29

the degrees of freedom, �, are calculated via moment matching. Specifically, for D0 =

diag(p0i(1− p0i)) and P0 = D0 −D0X(X′D0X)−1X′D0, we define �Q = tr(P0K)/2, I�� =

tr(KP0KP0)/2, I�� = tr(P0KP0)/2, I�� = tr(P0P′0)/2, and I�� = I�� − I2��/I��. Then

� can be estimated as � = I��/(2�Q) and we can calculate the p-value for significance

by comparing Q/� to a chi-square distribution of � degrees of freedom, �2� , where � =

2�2Q/I�� . The original derivation of our score test can be found in Lin,49 where the link

function in Equation 2 of Lin is assumed to be the logit and the design matrix (Z) is set

to be K1/2. Our score statistic, Q, in Equation 4 is identical to the first term of the score

statistic, U , from Equation 8 of Lin (as D = 1 and Δ−1W = D since the logit link is a

canonical link).

Acknowledgements

This work was sponsored by NIH grants CA76404 (to X.L.) and HG003618 (to M.P.E.).

Web Resources

The URLs for data presented herein are as follows:

Online Mendelian Inheritance in Man (OMIM): http://www.ncbi.nlm.nih.gov/

Omim

HAPGEN program: http://www.stats.ox.ac.uk/˜marchini/software/gwas/

gwas.html

R-functions for the logistic kernel machine test: http://www.bios.unc.edu/˜mwu/

software/

30

References

[1] Hunter, D., Kraft, P., Jacobs, K., Cox, D., Yeager, M., Hankinson, S., Wacholder, S.,

Wang, Z., Welch, R., Hutchinson, A., et al. (2007). A genome-wide association study

identifies alleles in FGFR2 associated with risk of sporadic postmenopausal breast

cancer. Nature Genetics, 39, 870.

[2] Easton, D., Pooley, K., Dunning, A., Pharoah, P., Thompson, D., Ballinger, D., Struew-

ing, J., Morrison, J., Field, H., Luben, R., et al. (2007). Genome-wide association study

identifies novel breast cancer susceptibility loci. Nature, 447, 1087–1093.

[3] Yeager, M., Orr, N., Hayes, R., Jacobs, K., Kraft, P., Wacholder, S., Minichiello, M.,

Fearnhead, P., Yu, K., Chatterjee, N., et al. (2007). Genome-wide association study of

prostate cancer identifies a second risk locus at 8q24. Nature Genetics, 39, 645–649.

[4] Gudmundsson, J., Sulem, P., Manolescu, A., Amundadottir, L., Gudbjartsson, D.,

Helgason, A., Rafnar, T., Bergthorsson, J., Agnarsson, B., Baker, A., et al. (2007).

Genome-wide association study identifies a second prostate cancer susceptibility

variant at 8q24. Nature genetics, 39, 631–637.

[5] Thomas, G., Jacobs, K., Yeager, M., Kraft, P., Wacholder, S., Orr, N., Yu, K., Chatterjee,

N., Welch, R., Hutchinson, A., et al. (2008). Multiple loci identified in a genome-wide

association study of prostate cancer. Nature Genetics, 40, 310–315.

[6] Sladek, R., Rocheleau, G., Rung, J., Dina, C., Shen, L., Serre, D., Boutin, P., Vincent,

D., Belisle, A., Hadjadj, S., et al. (2007). A genome-wide association study identifies

novel risk loci for type 2 diabetes. Nature, 445, 881–885.

[7] Scott, L., Mohlke, K., Bonnycastle, L., Willer, C., Li, Y., Duren, W., Erdos, M., String-

ham, H., Chines, P., Jackson, A., et al. (2007). A genome-wide association study of

type 2 diabetes in Finns detects multiple susceptibility variants. Science, 316, 1341.

[8] Saxena, R., Voight, B., Lyssenko, V., Burtt, N., de Bakker, P., Chen, H., Roix, J.,

Kathiresan, S., Hirschhorn, J., Daly, M., et al. (2007). Genome-wide association anal-

ysis identifies loci for type 2 diabetes and triglyceride levels. Science, 316, 1331–1336.

[9] Kraft, P. and Cox, D. (2008). Study designs for genome-wide association studies.

Advances in genetics, 60, 465.

[10] Schaid, D., Rowland, C., Tines, D., Jacobson, R., and Poland, G. (2002). Score tests

for association between traits and haplotypes when linkage phase is ambiguous. The

American Journal of Human Genetics, 70, 425–434.

[11] Hunter, D. and Kraft, P. (2007). Drinking from the fire hose–statistical issues in

genomewide association studies. New England Journal of Medicine.

[12] Liu, D., Ghosh, D., and Lin, X. (2008). Estimation and testing for the effect of a

genetic pathway on a disease outcome using logistic kernel machine regression via

logistic mixed models. BMC Bioinformatics, 9.

[13] Kwee, L., Liu, D., Lin, X., Ghosh, D., and Epstein, M. (2008). A powerful and flexible

multilocus association test for quantitative traits. The American Journal of Human

Genetics, 82, 386–397.

[14] Schaid, D., McDonnell, S., Hebbring, S., Cunningham, J., and Thibodeau, S. (2005).

Nonparametric tests of association of multiple genes with human disease. The Amer-

ican Journal of Human Genetics, 76, 780–793.

[15] Wessel, J. and Schork, N. (2006). Generalized Genomic Distance Based Regression

Methodology for Multilocus Association Analysis. American Journal of Human Ge-

netics, 79, 792.

[16] Kanehisa, M. and Goto, S. (2000). KEGG: Kyoto encyclopedia of genes and genomes.

Nucleic acids research, 28, 27.

[17] Ashburner, M., Ball, C., Blake, J., Botstein, D., Butler, H., Cherry, J., Davis, A., Dolin-

ski, K., Dwight, S., Eppig, J., et al. (2000). Gene Ontology: tool for the unification of

biology. Nature genetics, 25, 25–29.

[18] McAuliffe, J., Pachter, L., and Jordan, M. (2004). Multiple-sequence functional anno-

tation and the generalized hidden Markov phylogeny. Bioinformatics, 20, 1850–1860.

[19] Barrett, J., Fry, B., Maller, J., and Daly, M. (2005). Haploview: analysis and visualiza-

tion of LD and haplotype maps. Bioinformatics, 21, 263–265.

[20] Cristianini, N. and Shawe-Taylor, J. (2000). An introduction to support Vector Ma-

chines: and other kernel-based learning methods. (Cambridge Univ Pr).

[21] Brown, M., Grundy, W., Lin, D., Cristianini, N., Sugnet, C., Furey, T., Ares, M., and

Haussler, D. (2000). Knowledge-based analysis of microarray gene expression data

by using support vector machines. Proceedings of the National Academy of Sciences,

97, 262–267.

[22] Kimeldorf, G. and Wahba, G. (1971). Some results on Tchebycheffian spline

functions(Tchebycheffian spline functions, solving Hermite-Birkhoff interpolation as

stochastic prediction and filtering). Journal of Mathematical Analysis and Applica-

tions, 33, 82–95.

[23] Lin, W. and Schaid, D. (2009). Power comparisons between similarity-based multilo-

cus association methods, logistic regression, and score tests for haplotypes. Genetic

epidemiology, 33, 183–97.

[24] Mukhopadhyay, I., Feingold, E., Weeks, D., and Thalamuthu, A. (2009). Associa-

tion tests using kernel-based measures of multi-locus genotype similarity between

individuals. Genetic Epidemiology.

[25] Zhang, D. and Lin, X. (2003). Hypothesis testing in semiparametric additive mixed

models. Biostatistics, 4, 57–74.

[26] Lin, D. (2005). An efficient Monte Carlo approach to assessing statistical significance

in genomic studies. Bioinformatics, 21, 781–787.

[27] Cheverud, J. (2001). A simple correction for multiple comparisons in interval map-

ping genome scans. Heredity, 87, 52–58.

[28] Nyholt, D. (2004). A simple correction for multiple testing for single-nucleotide

polymorphisms in linkage disequilibrium with each other. The American Journal of

Human Genetics, 74, 765–769.

[29] Moskvina, V. and Schmidt, K. (2008). On multiple-testing correction in genome-wide

association studies. Genetic Epidemiology, 32.

[30] Hoh, J. and Ott, J. (2003). Mathematical multi-locus approaches to localizing complex

human trait genes. Nature Reviews Genetics, 4, 701–709.

[31] Zaykin, D., Westfall, P., Young, S., Karnoub, M., Wagner, M., Ehm, M., and Inc, G.

(2002). Testing association of statistically inferred haplotypes with discrete and con-

tinuous traits in samples of unrelated individuals. Hum Hered, 53, 79–91.

[32] Chapman, J., Cooper, J., Todd, J., and Clayton, D. (2003). Detecting disease associ-

ations due to linkage disequilibrium using haplotype tags: a class of tests and the

determinants of statistical power. Hum Hered, 56, 18–31.

[33] Roeder, K., Bacanu, S., Sonpar, V., Zhang, X., and Devlin, B. (2005). Analysis of

single-locus tests to detect gene/disease associations. Genetic epidemiology, 28, 207–

219.

[34] Tzeng, J. and Zhang, D. (2007). Haplotype-based association analysis via variance-

components score test. The American Journal of Human Genetics, 81, 927–938.

[35] Minichiello, M. and Durbin, R. (2006). Mapping trait loci by use of inferred ancestral

recombination graphs. The American Journal of Human Genetics, 79, 910–922.

[36] Tachmazidou, I., Verzilli, C., and De Iorio, M. (2007). Genetic association mapping

via evolution-based clustering of haplotypes. PLoS Genet, 3, e111.

[37] Saad, A., Meacham, W., Bai, A., Anelli, V., Elojeimy, S., Mahdy, A., Turner, L., Cheng,

J., Bielawska, A., Bielawski, J., et al. (2007). The functional effects of acid cerami-

dase overexpression in prostate cancer progression and resistance to chemotherapy.

Cancer biology & therapy, 6, 1455.

[38] Li, C., Park, J., He, X., Levy, B., Chen, F., Arai, K., Adler, D., Disteche, C., Koch, J.,

Sandhoff, K., et al. (1999). The human acid ceramidase gene (ASAH): structure,

chromosomal location, mutation analysis, and expression. Genomics, 62, 223–231.

[39] Spencer, C., Su, Z., Donnelly, P., and Marchini, J. (2009). Designing genome-wide

association studies: sample size, power, imputation, and the choice of genotyping

chip. PLoS Genetics, 5.

[40] Altschuler, D., Brooks, L., Chakravarti, A., Collins, F., Daly, M., and Donnelly, P.

(2005). International HapMap Consortium. A haplotype map of the human genome.

Nature, 437, 1299–320.

[41] Marchini, J., Howie, B., Myers, S., McVean, G., and Donnelly, P. (2007). A new mul-

tipoint method for genome-wide association studies by imputation of genotypes.

Nature genetics, 39, 906–913.

[42] Gao, X., Starmer, J., and Martin, E. (2008). A multiple testing correction method

for genetic association studies using correlated single nucleotide polymorphisms.

Genetic Epidemiology, 32.

[43] Price, A., Patterson, N., Plenge, R., Weinblatt, M., Shadick, N., and Reich, D. (2006).

Principal components analysis corrects for stratification in genome-wide association

studies. Nature genetics, 38, 904–909.

[44] Tibshirani, R. (1996). Regression shrinkage and selection via the lasso. Journal of the

Royal Statistical Society. Series B (Methodological), 58, 267–288.

[45] Devlin, B., Roeder, K., and Wasserman, L. (2003). Analysis of multilocus models of

association. Genetic epidemiology, 25, 36–47.

[46] Croiseau, P. and Cordell, H. (2009). Analysis of North American Rheumatoid Arthri-

tis Consortium data using a penalized logistic regression approach. In BMC proceed-

ings, volume 3, BioMed Central Ltd, pp. S61.

[47] Szymczak, S., Biernacka, J., Cordell, H., Gonzalez-Recio, O., K

”onig, I., Zhang, H., and Sun, Y. (2009). Machine learning in genome-wide associa-

tion studies. Genetic Epidemiology, 33, S51–S57.

[48] Zhang, D. and Lin, X. (2003). Hypothesis testing in semiparametric additive mixed

models. Biostatistics, 4, 57–74.

[49] Lin, X. (1997). Variance component testing in generalised linear models with random

effects. Biometrika, 84, 309–326.

[50] Satterthwaite, F. (1946). An approximate distribution of estimates of variance com-

ponents. Biometrics Bulletin, 2, 110–14.

Table 1: Empirical type-I error rates at �=0.05 for the logistic kernel machine test andindividual SNP analysis when applied to SNP sets simulated from the ASAH1 gene.

Individual Logististic Kernel Machine Testn SNP Analysis Linear Kernel IBS Kernel Weighted IBS Kernel

1000 0.049 0.052 0.046 0.0552000 0.048 0.047 0.053 0.052

Table 2: Top results from the discovery phase of the CGEMS breast cancer GWAS.SNP Chromosome Gene p-valuers10510126 10 2.0×10−6

rs12505080 4 8.0×10−6

rs17157903 7 RELN 9.0×10−6

rs1219648 10 FGFR2 1.2×10−5

rs7696175 4 TLR1—TLR6 1.4×10−5

rs2420946 10 FGFR2 1.5×10−5

rs2107349 7 AZGP1—AZGP1P2 1.7×10−5

rs6497337 16 SYT17 2.0×10−5

rs1250255 2 FN1 3.4×10−5

rs10804287 2 3.8×10−5

Table 3: Top results from the logistic kernel machine based SNP set analysis of the CGEMsBreast Cancer Study data.

Linear IBS Weighted IBSGene p-value q-value p-value q-value p-value q-valueFGFR2 7.69×10−7 0.01 2.53×10−6 0.03 1.35×10−5 0.05CNGA3 5.59×10−6 0.05 4.65×10−6 0.03 3.25×10−6 0.02TBK1 1.30×10−5 0.07 3.28×10−6 0.03 5.48×10−6 0.02VWA3B 1.53×10−5 0.07 7.84×10−6 0.03 3.99×10−6 0.02PTCD3 5.50×10−5 0.20 9.02×10−6 0.03 3.78×10−6 0.02XPOT 6.60×10−5 0.20 3.48×10−5 0.09 4.91×10−5 0.11VAPB 9.79×10−5 0.22 4.51×10−5 0.10 8.11×10−5 0.14SHC3 1.01×10−4 0.22 3.77×10−4 0.34 1.61×10−3 0.46SFTPB 1.78×10−4 0.31 1.38×10−4 0.27 7.62×10−5 0.14SPATA7 1.90×10−4 0.31 1.76×10−4 0.28 1.39×10−4 0.22

Figure 1: Empirical Power for SNP sets based on ASAH1 and LD-plot for the 86 SNPs inthe ASAH1 gene based on the CEU sample from the International HapMap Project. Thetyped SNPs are denoted with a triangle and the bottom panel shows the LD-structure ofthe SNPs in the ASAH1 gene.

Figure 2: Empirical Power for SNP sets based on ASAH1. The SNPs on the x-axis aresorted by median R2 with the 14 typed SNPs which is shown in the bottom plot.

1 3 5 7 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84

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Figure 3: Smoothed Empirical Power Curves as a function of mediaR2 between the causalSNP and the typed SNP for SNP sets based on a range of genes.

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Kernel Machine: n = 1000Indiv. Analysis: n = 1000Kernel Machine: n = 2000Indiv. Analysis: n = 2000

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Figure 4: Comparison of the power and type I error of the Logistic Kernel Machine Test,the Wessel and Schork method, and Mukhopadhyay et al.’s approach, denoted by K, W,and M respectively, under seven models. Power and size estimates are based on 500 and1000 simulations, respectively. The blue line shows the expected type I error rate.

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Power Type I Error